Carathéodory's criterion
Encyclopedia
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician
Constantin Carathéodory
. Its statement is as follows: Let
denote the Lebesgue
outer measure
on
, and let 
. Then 
is Lebesgue measurable if and only if 
for every 
.
Notice that
is not required to be a measurable set.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Constantin Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
. Its statement is as follows: Let
denote the LebesgueLebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
outer measure
Outer measure
In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by...
on
, and let . Then is Lebesgue measurable if and only if for every .Notice that
is not required to be a measurable set.